68 JAiyiES BERNOULLI. 



metical mean between 1 and 6. Again A has a chance equal to - 



of throwing a two at his first trial ; in this case he has two throws 

 for the stake, and these two throws may give him any number 

 between 2 and 12 inclusive; and the probability of the number 

 2 is the same as that of 12, the probability of 3 is the same as 



that of 11, and so on; hence as before we may take ^ (2 + 12), 



that is 7, for his mean throw. In a similar way if three, four, 

 five, or six be thrown at the first trial, the corresponding means 

 of the numbers in the throws for the stake will be respectively 

 lOi, 14i, 17^, and 21. Hence the mean of all the numbers is 



^ m + 7 + lOi + 1-i + I7i + 21], that is 121; 



and as this number is greater than 12 it might appear that the 

 odds are in favour of A. 



A false solution of a problem will generally appear more plau- 

 sible to a person who has originally been deceived by it than to 

 another person who has not seen it until after he has studied the 

 accurate solution. To some persons James Bernoulli's false solu- 

 tion 'would appear simply false and not plausible; it leaves the 

 problem proposed and substitutes another which is entirely differ- 

 ent. This may be easily seen by taking a simple example. 

 Suppose that A instead of an equal chance for any number of 

 throws between one and six inclusive, is restricted to one or six 

 throws, and that each of these two cases is equally" likely. Then, 



as before, we may take -^ (8 J + 21], that is 12J as the mean 



throw. But it is obvious that the odds are against him; for if 

 he has only one throw he cannot obtain 12, and if he has six 

 throws he will not necessarily obtain 12. The question is not 

 what is the mean number he will obtain, but how many throws 

 will give him 12 or more, and how many will give him less than 12. 

 James Bernoulli seems not to have been able to make out 

 more than that the second solution must be false because the first 

 is unassailable; for after saying that from the second solution we 

 might suppose the odds to be in fiiv^our of A, he adds, Hujus 



